Quantum Fourier Transform: Phasor Sum (2D)

Qubits n 3 (N=8)

Input State

Basis index k 0

Second index k₂ 4

Hidden period r 2

Transform

Scrub Output State

Output index j 0

c₀ built from N chained phasors

Dimension N

ΣP (unitarity)

1.000

|c_j| this j

0.354

What this 2D view adds

The 3D scene renders only the result: stem height = |c_j|, arrow = phase, one needle per output state. This 2D companion instead draws the arithmetic that produces that result — the QFT is literally a sum of rotated vectors:

QFT|ψ⟩_j = (1/√N) Σ_{x=0}^{N-1} a_x · e^{ dir·2πi·j·x/N }

Each term a_x·e^{iθ} is ONE phasor.
The left canvas chains all N of them
tip-to-tail; the resultant arrow (bold)
is the actual amplitude c_j.

Scrub the output index j slider (or hit Sweep all j) and watch the chain re-angle itself for every output state while the spectrum canvas below fills in bar by bar — the two views are two projections of the exact same sum.

Single |k⟩ — one input term; for any j the chain is N copies of a single unit-length step rotated by 2πjk/N per term, so magnitude is always exactly 1/√N (the 3D sim's case).
|k₁⟩+|k₂⟩ — superposition of two basis states; only two of the N terms are non-zero, so the "chain" is just two phasors added — magnitude now depends on j and can interfere constructively or destructively.
Period comb — input amplitude sits only on every r-th basis state (period r inside the register). This is the exact post-measurement state Shor's algorithm produces before its QFT step: the transform concentrates all output probability onto multiples of N/r, and the recovered period is read directly off the peak spacing.

Real-world relevance: Shor's factoring algorithm works by modular exponentiation collapsing the input register into a period-r comb like the third mode here, then applying exactly this QFT so the period becomes a measurable spacing in the output distribution — the mechanism visualized directly by the "Period comb" mode.

Phasor chain — complex plane (Σ for output j)

Chain segment (one input term) Resultant c_j

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Output spectrum |c_j| (hue = phase)